Yep, you need a polarizer. Now for the unsolicited advice (you knew it was coming...):

Figure out what the filter diameter of the largest lens you may ever use on your camera is and buy a polarizer at least that big. Use step-up rings to use it on smaller diameter lenses. that way you only have to buy one (they are rather expensive) polarizer.

For example, I have a Nikon Coolpix 4500 that uses 28mm filters. I have enough step-up rings that I can use my 72mm polarizer on it, as well as using it on all my 35mm lenses (except the new 28-85 zoom, which takes 82mm filters- oops!).